It is in my opinion definitely pride that makes Achilles refuse to fight when Briseis is taken.
First of all, there is not really so much of a tradition of romantic love back in those days in Greece. I do not really think that Achilles would have fallen in love with Briseis. This is even more so since she was just a prize of war.
Second, I think that one major theme of Homer's works is that honor was everything for a Greek warrior in these times. Achilles, seeing himself as the greatest warrior, would surely have felt his honor was being slighted when his prize was taken.